The history of projective geometry is a very complex one. Most of
the more formal developments on the subject were made in the 19th
century as a result of the movement away from the geometry of Euclid.
If one digs a little deeper, however, one can see th at the basic
concepts upon which this branch of geometry is based can be traced
back as far as the 4th century. These very early discoveries along
with Euclid's Elements are the building blocks for the
foundations that were laid down by the projective geometers of the
17th century. It is here that the history of the subject
becomes most interesting.
Great strides were made in the 17th century, but for some reason
projective geometry did not become popular among mathematicians until
the 19th century. Let us examine th e developments that were made in
the 17th century, what things led up these discoveries, and the
causes for the hundred-year lapse between these discoveries and their
further development in the 19th century.

What exactly is projective geometry? On the surface, it is simply
the study of the "properties of figures which remain unaltered
(invariant) in projection." (Lanczos, 280) What is meant by
projection? A very simple example is as follows. Look at a checke r
board head on. All of the lines are parallel. Turn that same board
at an angle keeping your perspective the same and what you see is
quite different. The lines are no longer parallel. From a
geometrical standpoint, what you are seeing is a projection of
the lines of the checkerboard onto another plane. Projective
geometry is the study of the properties of these lines after they have
been projected. Early projective geometers found that while lengths,
areas and angles were not maintained, there were prop erties of points
and lines which were unaffected or invariant in projection. Using
these discoveries they were able to construct new ways to solve old
problems, and that is how this now highly regarded genre of geometry
arose.

As defined above, projective geometry is the study of invariant
properties. "The earliest projective invariant is a cross ratio of
four collinear points." (Coolidge, 88) The cross ratio is a
fundamental quantity in projective geometry, and is comparable to
the notion of "distance" in traditional geometry.
The distance between two points is
a numerical value which
describes a
meaningful relationship between those points,
because "distance" is invariant in metrical geometry. However, in the
realm or projective geometry, this quantity does not remain invariant
and is therefore useless. The cross ratio is an alternative which
describes a relationship among four collinear points and also remains
unchanged in projection. (Lanczos, 287)

If four collinear points A, B, C, D are assigned the values
(a),(b), (a + ub), (a + vb)
respectively, then the cross ratio is defined
as either v/u or u/v. (Coxeter, 177)
The proof that this quantity is
invariant is more complicated.
The figure below shows the points A, B and C on the
line l being projected onto the
points A', B', and C' on the line
l' from the perspective point O.
We also see a series of points along
the X axis labeled x, x'
or x.

A summary of the proof that this figure illustrates is as follows.

The x's are expressed in terms of their corresponding
x
values. CA,
CB, C'A', and C'B'
are represented by (x3 - x1),
(x3 - x2),
(x'3 - x'1)
and (x'3 - x'2) respectively.
The ratio CA/CB simplifies to a
product in which the "position of the point C drops out"
(Lanczos, 288).
Then a similar result is produced by C'A'/C'B'.
This is important because then the ratio CA/CB : C'A'/C'B' is equal to a
function of x1 and
x2.
Because the position of the point C cancelled
out, we can then say that: CA/CB : C'A'/C'B'= DA/DB : D'A'/D'B'. This
can then be rewritten though properties of ratios as:CA/CB : DA/DB = C'A'/C'B' : D'A'/D'B'.
Then we recognize that
CA/CB : DA/DB is in fact the cross ratio.
We can then rewrite this in
its more common notation; (AB, CD).
From this we have: (AB, CD) = (A'B', C'D').
Therefore the cross ratio is invariant under
projection. It is uncertain who exactly is responsible for the
discovery of the invariance of the cross ratio, but it is known that
Pappus of Alexandria (290-350), who is considered by some to be the
discoverer of the "earliest truly projective theorems." (Coxeter, 3) ,
at least knew of its existence. Whether he in fact discovered it is
not certain. Pappus wrote a commentary on Euclid's lost book of
porisms and it is in that commentary that the earliest recording of
the cross ratio invariant is known. It is suspected h owever that
Euclid himself may have been aware of the invariant. (Coolidge, 89)

Let us now examine
the Desargues Configuration:

The Desargues Configuration is a simple yet intriguing example of
the use of projection. What we see here is the projection of the
triangle ABC onto a triangle A'B'C' in another pl ane.
To understand this figure, it must be thought of it in three
dimensions. The point O is the perspective point. An observer
at the point O would not see both triangles. From that point,
the triangle A'B'C' is exactly hidden behind ABC. Call
the plane in which ABC lies P1, and the plane
in which A'B'C' lies P2. The line AB lies in
P1 while A'B' lies in P2 but we
know they intersect because they both lie in the plane B'OA'. The
point at which they intersect is called C''. This point C''
must lie
on the line of intersection of the planes P1 and
P2. By similar logic, the points B'' and A''
are marked as the intersections of lines AC with A'C'
and of BC with B'C'. Just like C'',
these points must lie on the line of intersection of the two planes.
Thus all three lie on a single straight line, and they are
collinear.
What is even more fascinating about this configuration is
that of the ten points in this figure, any one of them can be chosen
as the point of perspective. With a few careful manipulations, this
configuration can be reconstructed to work in the same way from the
perspective of any one of the ten points. (Lanczos, 281)

This figure was named after the French mathematician Gérard
Desargues (1591 - 1661) , who is considered by many to be the true
founder of Projective Geometry. Desargues was an engineer and
architect who became interested in the concept of projection. He was
not however interested in projection on a strictly mathematical basis.
Rather he expressed an interest in the education of artists and
engineers (Kline, 288). He gave some lectures on the subject in Paris
between the years 1626 and 1630. (Ball, 25 9) His most notable work,
however was the Brouillon projet which was published in 1639. This
essay dealt with conics. It was around Desargues' time that
mathematicians were beginning to feel that Greek methods of proof were
weak. These methods were not applicable in general, but rather were
specific to each theorem. (Kline, 286) There was a desire to develop
some uniformity in methods of proof. Desargues had studied the work
of Apollonius and took an interest in simplifying his ideas of conics.
He dev eloped new ways to prove many theorems on conics through the
use of projection. Desargues published his ideas in his Brouillon
projet of which he made only fifty copies, which he distributed among
friends. Fermat, one of Desargues' contemporaries, was hi ghly
impressed with the book and "regarded Desargues as the real founder of
the theory of conic sections." (Kline, 289) Though Fermat's reaction
was extremely positive, that appreciation was not shared in general
among the mathematics community of the tim e. As a result of the lack
of enthusiasm Desargues retired. It would not be for another hundred
years that the brilliance of his work would be recognized.

Desargues was not the sole projective geometer of the 17th century.
Two other mathematicians credited with work on the subject were Blaise
Pascal and Philippe de La Hire. Pascal, influenced by Desargues, took
a strong interest in projective geometry. He focused on simplifying
the properties of conic sections. Pascal produced an essay that
unfortunately was lost but was read by Leibniz who called it, "so
brilliant that he could not believe it was written by so young a man."
(Kline, 297) Pascal's is most n otable work in the field of projective
geometry bears his name. Pascal's Theorem is as follows; "If a
hexagon is inscribed in a conic, the three points of intersection of
the pairs of opposite sides lie on one line." (Kline, 297) Though
Pascal's exact proof of this theorem is not known (Kline, 297),
the figure below illustrates one of many proofs which have since been
proposed.

The specifics of this proof incorporate aspects of
projective geometry which require more explanation than is convenient
here.
To summarize, the leftmost of the two circles is the "conic"
in which the hexagon AB'CA'BC' is inscribed. L, M, and N are the
three collinear points. Side BC' intersects with B'C at the point L,
AC' with A'C at M, and AB' with A'B at N. The smaller circle to the
ri ght is a projection of the conic. In this smaller circle, various
projective properties are used to show all sides of the triangle JKM
parallel to the corresponding sides of triangle BB'L. By a converse
theorem, JB, KB', ML are parallel. "Parallel" has a very different
meaning in projective geometry. Keep in mind that lines resulting
from the projection of two parallel lines are not assumed to be
parallel as well. In this figure, JB, KB' and ML intersect. Proving
them "parallel" is simply one of several s teps in the projective
reasoning which ultimately brings us to the conclusion that ML passes
through the point at which JB intersects KB' which is the same point
at which A'B intersects AB'. This point is also equal to N, which
concludes the proof.

Philippe de la Hire was also heavily influenced by Desargues and
strongly interested in projective geometry. He is most noted for his
work, Sectiones Conicae ("Conic Sections").
This work dealt entirely with projective
geometry. La Hire believed strongly that the methods
of projection were stronger by far than Appollonios' methods. He
thus attempted to prove all 364 of Appolonios' theorems. He came very
close to this goal, proving 300 of them. (Kline, 298)

It is quite intriguing, when looking at the history of projective
geometry, to see that following the many great advances of Desargues,
Pascal, and La Hire, more than a hundred years pass during which the
subject is virtually untouched. The work of Desargues was actually
not appreciated at all outside the circle of his friends and colleagues at that
time. (Kline, 289) What accounts for this lack of appreciation and
why was it not until the early 19th century that this subject becomes
popular? Essentially, it is said
that Descartes' work on analytic geometry in the 18th century drew
attention away from the "pure mathematics" of projection and towards
more analytic genres. There are many theories as to why this
occurred.

One thought is that Desargues' work lacked clarity. How can people
appreciate something they cannot fully understand? Cynthia Cook stated
in Western Mathematics Comes of Age that the ideas of Desargues were
not popular among mathematicians of the time because while his
definitions "helped to organize the rules of perspective for artists
and craftsmen[,] ... their philosophical meaning was unclear."
(Cook, 95) If his work appeared to some as simply a catalogue of the rules
of perspective then that could be
a legitimate explanation for the lack of enthusiasm in the
mathematics community. In addition to this lack of apparent
mathematical significance, Desargues also used some terminology that
may have baffled his readers. For instance, he used the word
"palm" to describe a straight line. Some of the words he used can also
be found in earlier work of Alberti, but they were not in common
use. It is the opinion of some that his choice of language made it
difficult to understand the Brouillon Projet (Kline, 289). Could it
be that Desargues' work, however brilliant, was so unclear as to delay
the progress of projective geometry for a century? Let us explore other
possible contributing factors.

It is also suggested that the interest of mathematicians of the 17th
century involved discoveries which were applicable to science and
technology and "[a]lgebraic methods of working with mathematical
problems proved in general to be more effective . . ." w hile, ". .
.[t]he qualitative results the projective geometers produced by their
synthetic methods were not nearly so helpful." (Kline, 301) Analytic
geometry was becoming increasingly more popular and historians believe
that the reason for that is simply
because it was more applicable to the needs of the time. There was
an interest at this time in the use of mathematics to describe natural
phenomena like motion, heat, and pressure. Analytic geometry allowed
for the construction of curves to represent dat a which was much more
conducive to creating "laws" of nature. "The interaction between the
two modes of analysis (algebra and geometry) provided fresh insights
into the nature of physical reality."(Cook, 97) Perhaps the clarity of
Desargues' work was irre levant and it was inevitable that the birth
of Analytic geometry would cast a temporary shadow on projective
geometry.

Another thought is that perhaps projective geometry of the 17th and
19th centuries are more different than one would think and it is thus,
not unusual for them to be separated by so many years. The difference
is in the direction and intent of the geomete rs. As discussed
earlier, the intention of Desargues and other projective geometers of
his time was to better prove the methods of Euclid. They were "far
from thinking in terms of new geometry." (Kline, 300) They were not
thinking, as others of the time w ere, of advancement along the lines
of "new geometry," but rather in terms of improving upon what already
existed. However, Rouse Ball in his work, A Short Account of the
History of Mathematics, states that in the 17th century, "Greek
mathematics was not capable of much further extension, mathematicians
were beginning to seek new methods of geometry." (Ball, 258) In the
19th century, projective geometry is reborn as a result of the
development of several new non-Euclidean branches of geometry.
Perhaps it was necessary to view projection as "new" in order to make
advancements.

We have explored several possible causes for the initial neglect of
projective methods in the 17th century, as contrasted with its rebirth
in the 19th century, and while these potential causes differ
considerably, one can detect a common theme. Mathematics was simply
not ready for projective geometry in the 17th century. Analytic
geometry had to come first. Perhaps Desargues was before his time, or
perhaps he didn't even see projective geometry the way it would
eventually come to be seen in the 19th centur y; as an independent
branch. Regardless, the subject was eventually reborn and historians
now refer to the ideas of Desargues as "fundamental to our
understanding of space and time." (Cook, 95)

BIBLIOGRAPHY

Ball, W. W. Rouse, A Short Account of the History of Mathematics,
Macmillan and Co., London. 1912.